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Abstract
We theoretically investigate the linear susceptibility and propagation of light in a three-dimensional (3-D) Rydberg gas under conditions of electromagnetically induced transparency. Rydberg atoms with two relevant S states are coupled via exchange interactions. When the gas is initially prepared in an entangled spin-wave state, this coupling induces a strong, nonlocal susceptibility whereby the photon field at one point of the medium acts as a source at a distant position. The nonlocal propagation occurs not only in the propagation direction but also in the paraxial direction. We discuss the absorption features and numerically simulate the 3-D propagation of probe laser light. Combined with the long-range exchange interaction, we show that the 3-D Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and sign of the nonlocal interaction kernel can be widely tuned.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Gases of interacting Rydberg atoms [1,2] have recently attracted theoretical [3–9] and experimental [10–15] attention. The atoms in a Rydberg gas undergo long-range dipole-dipole interactions within a certain blockade distance [16–18]. Under conditions of electromagnetically induced transparency (EIT) [19], strong atom-light interactions occur without absorption, and we can probe novel quantum phenomena such as many-body phenomena with strongly interacting photons [20,21], and nonlinear and nonlocal light-matter interactions [22–25]. Here, a physical mechanism is an essentially classic density-density interaction between the Rydberg states [26]. However, in some cases, these systems may exhibit resonant dipole-dipole interactions [27] and consequent excitation exchanges [28,29]. The exchange interaction establishes coherence between distant atoms, which leads to the linear susceptibility of the medium acquiring a nonlocal term, and the system’s (nonlocal) optical response is dramatically altered.
The authors of Ref. [30] have studied a Rydberg-atom system with exchange interactions initially prepared in a spin-wave state [31,32]. After discussing the underlying mechanism in a two-atom model, they have numerically investigated a one-dimensional atomic gas. Here, we expand the one-dimensional atomic gas model to a three-dimensional (3-D) ensemble of atoms in a spatially homogeneous spin wave and analytically discuss the linear optical response of this strongly interacting Rydberg-EIT medium. The exchange interactions between Rydberg-state atoms induce strong nonlocal interactions between photons, whereby the photon field at one point of the medium can act as a source at a distant position. The "nonlocal" propagation occurs not only in the propagation direction, as discussed in [30], but also in the paraxial direction. We discuss the absorption features, particularly the interplay of the one-photon detuning and phase-matching relations of the involved light fields. Combined with the long range exchange interaction, the Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and the sign of the nonlocal interaction kernel can be widely tuned. To demonstrate the potential of this medium, we numerically simulate the propagation of probe laser light. Our results indicate that the medium enhances the absorptive properties of Rydberg gases.
The remainder of this paper is structured as follows. Section 2 introduces a generic interacting many-body system in the EIT configuration. Section 3 discusses the optical response of the 3-D system and derives linear susceptibility in k space. The effective potential is investigated, and light propagation is numerically simulated in a 3-D gas. The main results and conclusions are summarized in Section 4.
2. System
We consider a beam with wave number $\mathbf {k}_p$ and amplitude $E_p(\mathbf {r},t)$ passing through an atomic medium. Figure 1 depicts the level structure of the medium, in which $|1\rangle$ and $|2\rangle$ are the ground state and low-lying excited state, respectively, and the upper levels $|3\rangle$ and $|4\rangle$ are Rydberg S states. The transition between $|1\rangle$ and $|2\rangle$ is driven by a weak probe field with detuning $\delta$ and a Rabi frequency of $\Omega _p(\mathbf {r},t) = \mu _{21}E(\mathbf {r},t)\exp [i(\mathbf {k}_p\cdot \mathbf {r})]$ where $\mu _{21}$ is the transition dipole moment. The transition between $|2\rangle$ and $|3\rangle$ is coupled by a strong control laser field with Rabi frequency $\Omega _c(\mathbf {r})=\Omega _c \exp [i(\mathbf {k}_c \cdot \mathbf {r})]$. Together, these transitions induce a typical EIT configuration. In the dipole and rotating wave approximation, the corresponding single-particle Hamiltonian is
The density matrix $\rho$ of the system is evolves under the following quantum master equation:
where $\mathcal {D}(\sigma )\rho = \sigma \rho \sigma ^{\dagger } - \frac {1}{2}\{\sigma ^{\dagger }\sigma , \rho \}$ is the dissipator, $H = \sum _{j=1}^{N} H_j + \sum _{j>k}^N (V_{jk}^d + V_{jk}^e)$, and $\gamma$ is the dissipative decay of the short-lived state $|2\rangle$ into the ground state $|1\rangle$. The decay from Rydberg states $|3\rangle$ and $|4\rangle$ is neglected.3. Light propagation in a 3-D gas
3.1 Optical response of a gas
We first consider a two-atom system initially existing in a spin-wave or super-atom state $|\Psi (\mathbf {r}_1,\mathbf {r}_2)\rangle = (|14\rangle + e^{i\mathbf {k}_s\cdot \mathbf {r}_{12}}|41\rangle )/\sqrt 2$, where $\mathbf {r}_{12} = \mathbf {r}_1 - \mathbf {r}_2$. The system will evolve in the two-atom basis $\{|14\rangle , |24\rangle , |41\rangle , |42\rangle , |34\rangle , |43\rangle \}\equiv \{|1\rangle , |2\rangle , |3\rangle , |4\rangle , |5\rangle , |6\rangle \}$. The optical response of atom $i$ positioned at $\mathbf {r}_i$, $\langle \sigma _{21}(\mathbf {r}_i)\rangle = \mathrm {Tr} [\rho \sigma _{21}^i]$, is given by the coherence between states $|1\rangle$ and $|2\rangle$. To capture the feature of the optical response, we calculate the susceptibility using the first-order perturbation method, considering the first-order coupling in terms of $\Omega _p$. We assume $\rho \simeq \rho _0 + \rho _1$, where $\rho _0$ and $\rho _1$ are the zeroth-order and first-order corrections of the density matrix, respectively. When $\Omega _c \gg \Omega _p$, our initial state $|\Psi (\mathbf {r}_1,\mathbf {r}_2)\rangle$ is a metastable state in the metastable decoherence-free subspace [29,30]. We assume $\rho _0 = |\Psi (\mathbf {r}_1,\mathbf {r}_2)\rangle \langle \Psi (\mathbf {r}_1,\mathbf {r}_2)|$, i.e., the nonzero matrix elements in $\rho _0$ are constant and the other matrix elements (coupling the initial state) evolve over time. The two-atom Hamiltonian can be written as $H = H_0 + H_1$, where $H_0 = \sum _{j=1,2} [\delta (|2\rangle _j \langle 2| + |3\rangle _j \langle 3|) + \frac {\Omega _c(\mathbf {r}_j)}{2}\sigma _{32}^{j} + h.c. ] + V_{12}^d + V_{12}^e$ excludes the probe field, and $H_1 = \sum _{j=1,2}[\frac {\Omega _p(\mathbf {r}_j)}{2}\sigma _{21}^j + h.c.]$. The master equation for $\rho _1$ is then given by
Here we are interested in a three-dimensional ensemble of atoms. The atoms are in a spatially homogeneous spin wave, i.e., $|\Psi (\mathbf {r}_1,\ldots ,\mathbf {r}_N)\rangle = \sum _{j}c_j e^{i\mathbf {k}_s\cdot \mathbf {r}_j}|1\ldots 4_j \ldots 1\rangle$, with the probability amplitudes $c_j$ and the spin-wave vector $\mathbf {k}_s$. In the continuum limit, we set continuous densities $n(\mathbf {r})$, and normalized them by $\int n(\mathbf {r})d \mathbf {r} = 1$, thus obtain
Unlike the optical response in conventional EIT, the coherence in Eq. (8) involves a nonlocal term describing the dependence of an atoms response not only on the local field $E(\mathbf {r}_i)$, but also on the electric fields at more distant locations. As shown in Ref. [30], the nonlocal term emerges through the exchange interaction and initial entangled state. Note that if the exchange interaction between the atoms is absent, then $A_{+} = A_-$ so $\alpha _n=0$.
The propagation of the probe field is governed by the Maxwell equation. Under the slowly varying amplitude approximation, this equation reads
The second term on the left-hand side of Eq. (9) describes the diffraction effect of the probe field. For convenience and without loss of generality, we assume a long time duration of the probe field. The system then works in a steady state, and the time derivative in the Maxwell Eq. (9) can be neglected (i.e., $\partial /\partial t = 0$) [33,34]. The wave equation in Eq. (9) is then written as
3.2 Susceptibility in $k$ space
To understand the basic propagation physics, it is convenient to compute the susceptibility (local part $\chi _l (\mathbf {r})$ and nonlocal part $\chi _n (\mathbf {r} - \mathbf {r'})$) in $k$ space through the following Fourier transform (see Supplement 1):
In Figs. 2(a) and 2(b), we plot the nonlocal susceptibility $\tilde \chi _n (\mathbf {k})$ as function of $\mathbf {k}$ in the absence and presence of detuning $\delta$. It shows that the nonlocal susceptibility depends on both the wave number $\mathbf {k}$ and the detuning $\delta$. Its imaginary part goes negative at certain values of k, where the probe field gains strength. To show this feature, we consider a Gaussian probe field pulse in $\mathbf {k}$ space and plot the $k_z$-integrated of the probe field $\mathcal {E}(k_x,k_y)=\int \mathrm {Im}[\tilde \chi _n(\mathbf {k})]\tilde E(\mathbf {k})dk_z/ \int \tilde E(\mathbf {k})dk_z$. For comparison, we consider the phase-matched case ($\mathbf {K} = 0$) in the absence and presence of detuning $\delta$ (Figs. 2(c) and 2(d), respectively). We find that $\mathcal {E}(k_x,k_y)$ non-monotonically decays for a Gaussian probe field pulse in the $\mathbf {k}$ space. The increase of $\mathcal {E}(k_x,k_y)$ at some points directly results from the negative imaginary part of the nonlocal susceptibility and the resulting gain.
Fig. 2. (a-b) Real (dashed line) and imaginary (solid line) parts of the nonlocal susceptibility $\tilde \chi _n(\mathbf {k})$ for $\delta = 0$ (a) and $\delta = \gamma$ (b). (c-d) $\mathcal {E}(k_x,k_y)$ in the k-space when $\delta = 0$ (c) and $\delta = \gamma$ (d) in the phase-matched situation ($\mathbf {K} =0$). The other parameters are $\Omega _c=2\gamma , g= 0.1\gamma , L= 28.1\mu$m, $C_d=9.7\times 10^6\mathrm {GHz}\mu \mathrm {m}^6, C_e=7.5\times 10^6\mathrm {GHz}\mu \mathrm {m}^6$.
3.3 Light propagation through a 3-D gas
In the following we investigate the wave equation (13) on a space-time grid. Assuming that the spatial length of the probe pulse in the $z$ direction is much larger than the range of atom-atom interactions, we make a local approximation along the z direction [23,24,33]. The last term on the right-hand side of Eq. (13) then reduces to $\int \chi _n(\mathbf {r}-\mathbf {r'}) E(\mathbf {r}')d \mathbf {r}'\simeq \int \chi _n'(\mathbf {r}_\bot -\mathbf {r'}_\bot ) E(\mathbf {r}'_\bot ,z)d \mathbf {r}'_\bot$, with $\chi _n'(\mathbf {r}_\bot -\mathbf {r'}_\bot )=\int \chi _n(\mathbf {r}-\mathbf {r'})dz'$. We also define the scaled time $\tau = z/(k_p |R_b^+|^2)$, $\bar {\mathbf {r}}_\bot = \mathbf {r}_\bot /|R_b^+|$ and the dimensionless probe amplitude $\bar E$ normalized to $\int d\bar {\mathbf {r}_\bot } |\bar E(\bar {\mathbf {r}_\bot },\tau )|^2 = 1$. Assuming $\Phi _{\mathbf {rr'}}\approx 2\pi \sqrt {|\bar {\mathbf {r}}_\bot -\bar {\mathbf {r}}'_\bot )|^2+\bar z^2}$ and, for simplicity, a homogeneous atom density, we get
Fig. 3. (a-b) Local potential changes with $\delta$ and $r_\bot$ showing (a) the imaginary part and (b) the real part of the local potential. (c-d) Nonlocal potential changes with $\delta$ and $r_\bot$: (c) the imaginary part and (b) the real part of the nonlocal potential. The other parameters are listed in the caption of Fig. 2.
We then simulated the 3-D propagation of a Gaussian probe field pulse. We also assume a Gaussian distribution of atom densities. Figure 4 presents the calculated output intensity profiles for different detunings in the phase-matched case. The shape of the probe field changed throughout the propagation. In particular, the intensity increased at some points and decreased at other points. For example, when $\delta$ was 0 (no detuning), the intensity spread along the transverse direction forming a ring. When $\delta$ was $\delta =0.3\gamma$, the intensity was focused at the center of the transverse direction. When $\delta =-0.4\gamma$, the intensity was localized at certain points. In the phase-unmatched case (Fig. 5), the phase parameter comes from the nonlocal potential, and the differences between the plots reflect the effect of the nonlocal terms on the probe light propagation. The features of the light propagation strongly depended on the one-photon detuning and phase-matching relation among the involved light fields. Combined with the long-range exchange interaction, the 3-D Rydberg gas is an ideal medium for studying nonlocal wave phenomena, in which the strength, range, and sign of the nonlocal interaction kernel can be widely tuned.
Fig. 4. Intensity profiles in the phase-matched situation ( $\mathbf {K} =0$) for different detuning values: (a) $\delta =0$, (b) $\delta =0.3\gamma$, (c) $\delta =-0.4\gamma$ and $\bar z=0\ (1), \bar z=1/3 \bar L_z\ (2), \bar z=2/3 \bar L_z\ (3)\ \texttt {and}\ \bar z=\bar L_z\ (4)$, respectively. The other parameters are $\Omega _c=\gamma , g= 0.1\gamma , V_d=0.3\gamma , \bar L_z=3 |R_b^+|$. For color coding each distribution has been normalized by the actual maximum intensity $|E_{max}|^2$.
Fig. 5. (a1-f1): The Effective nonlocal potential ($\times 10^3$) in the phase-nonmatched situation with (a) $K_x=1, K_y=0$ (b) $K_x=K_y=1$, (c) $K_x=2, K_y=0$, (d) $K_x=K_y=2$, (e) $K_x=3, K_y=0$ and (f) $K_x=K_y=3$ with $\delta =-0.4\gamma$. (a2-f2) The corresponding compressed output intensity profile. The other parameters and the color coding is identical to Fig. 4.
4. Summary and conclusions
In this work, we studied nonlocal light 3-D propagation in Rydberg EIT with an initial spin-wave state. We derived analytical expressions for the optical susceptibility of the medium and showed its nonlocal and k-dependent properties, through which light absorption or transmission enhances at some positions. The discussed features are expected in other initial states such as multiple Rydberg excitations. As an interesting future problem, we could investigate the coupled collective dynamics of an ensemble [35] and a propagating probe field in Rydberg CEIT experiments utilizing two interacting Rydberg states [12,28,36].
Funding
National Natural Science Foundation of China (12074433, 11871472); National Key Basic Research Program For Youth (2016YFA0301903); Natural Science Foundation of Hunan Province (2018JJ2467).
Acknowledgments
The author would like to thank Weibin Li for the fruitful discussions.
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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